Astronomy. Astrophysics. Gamma-ray burst optical light-curve zoo: comparison with X-ray observations,

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1 DOI: / / c ESO 2013 Astronomy & Astrophysics Gamma-ray burst optical light-curve zoo: comparison with X-ray observations, E. Zaninoni 1,2,M.G.Bernardini 1, R. Margutti 3,S.Oates 4, and G. Chincarini 1,5 1 INAF Osservatorio Astronomico di Brera, via Bianchi 46, Merate, Italy elena.zaninoni@brera.inaf.it 2 University of Padova, Physics & Astronomy Dept. Galileo Galilei, vicolo dell Osservatorio, Padova, Italy 3 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 4 Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK 5 Univerisità Milano Bicocca, Dip. Fisica G. Occhialini, P.zza della Scienza 3, Milano, Italy Received 1 February 2013 / Accepted 13 March 2013 ABSTRACT Aims. We present a comprehensive analysis of the optical and X-ray light curves (LCs) and spectral energy distributions (SEDs) of a large sample of gamma-ray burst (GRB) afterglows to investigate the relationship between the optical and X-ray emission after the prompt phase. We consider all data available in the literature, which where obtained with different instruments. Methods. We collected the optical data from the literature and determined the shapes of the optical LCs. Then, using previously presented X-ray data, we modeled the optical/x-ray SEDs. We studied the SED parameter distributions and compared the optical and X-ray LC slopes and shapes. Results. The optical and X-ray spectra become softer as a function of time while the gas-to-dust ratios of GRBs are higher than the values calculated for the Milky Way and the Large and Small Magellanic Clouds. For 20% of the GRBs the difference between the optical and X-ray slopes is consistent with 0 or 1/4 within the uncertainties (we did it not consider the steep decay phase), while in the remaining 80% the optical and X-ray afterglows show significantly different temporal behaviors. Interestingly, we find an indication that the onset of the forward shock in the optical LCs (initial peaks or shallow phases) could be linked to the presence of the X-ray flares. Indeed, when X-ray flares are present during the steep decay, the optical LC initial peak or end plateau occurs during the steep decay; if instead the X-ray flares are absent or occur during the plateau, the optical initial peak or end plateau takes place during the X-ray plateau. Conclusions. The forward-shock model cannot explain all features of the optical (e.g. bumps, late re-brightenings) and X-ray (e.g. flares) LCs. However, the synchrotron model is a viable mechanism for GRBs at late times. In particular, we found a relationship between the presence of the X-ray flares and the shape of the optical LC that indicates a link between the prompt emission and the optical afterglow. Key words. gamma-ray burst: general radiation mechanisms: non-thermal 1. Introduction Gamma-ray bursts (GRBs) are the most powerful sources of electromagnetic radiation in the Universe, with an isotropic luminosity that can reach values of erg s 1.TheSwift satellite (Gehrels et al. 2004), launched in November 2004, opened a new era for the study and understanding of GRB phenomena; thanks to the rapid response of the instruments with a small field of view, it was discovered that both the X-ray light-curve (LC; e.g. Nousek et al. 2006; Zhang et al. 2006) and the optical LC (e.g. Roming et al. 2009) have a complex shape. The rapid computation of the GRB position by the Swift Burst Alert Telescope (BAT; Barthelmy et al. 2005), refined with an accuracy of few arcseconds by the Swift X-ray Telescope (XRT, Burrows et al. 2005), and the instantaneous dissemination to the community Appendix C is available in electronic form at Full Tables C.1C.6 are only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr ( ) orvia via the GCN 1 allows a growing number of robotic telescopes to promptly repoint to the source. Some examples are the Robotic Optical Transient Search Experiment (ROTSE-III, Akerlof et al. 2003), the Rapid Eye Mount telescope (REM, Zerbi et al. 2001; Chincarini et al. 2003), the Gamma-Ray Burst Optical/Near- Infrared detector (GROND, Greiner et al. 2008), Liverpool (LT) and Faulkes telescopes (Gomboc et al. 2006), Télescopes à Action Rapide pour les Objets Transitoires (TAROT, Klotz et al. 2008b), and others. Some generic features have been previously found in optical LCs. Optical and X-ray LCs are different at early times in the majority of cases (Melandri et al. 2008b; Rykoff et al. 2009; Oates et al. 2009, 2011). In particular, Oates et al. (2009, 2011) noted that the optical LCs can decay or rise before 500 s after the trigger in the observer frame and do not show the steep decay as the X-ray LCs; after 2000 s the optical and X-ray LCs have similar slopes. Panaitescu & Vestrand (2008, 2011) divided the optical LCs according to their initial behavior (peaky or shallow). Peaks were associated to impulsive ejecta releases, while 1 Gamma-ray Coordinates Network (GCN), nasa.gov/gcn3_archive.html Article published by EDP Sciences A12, page 1 of 55

2 plateau phases were assumed to belong to the energy released by a long-lived central engine. Chromatic and achromatic breaks have been found in the optical and X-ray LCs (Melandri et al. 2008b; Rykoff et al. 2009; Oates et al. 2009, 2011; Panaitescu & Vestrand 2011). Moreover, the brighter optical LCs decay faster (Oates et al. 2009, 2011, 2012). When optical and X-ray LCs do not share the same temporal decay, X-ray LCs have been found to decay faster (Oates et al. 2009, 2011; Panaitescu & Vestrand 2011). For only a few GRBs with shallow X-ray decay phases we find a corresponding shallow decay in the optical (Rykoff et al. 2009; Li et al. 2012). Flares can occasionally appear in optical LCs and likely linked to the long term central engine activity (Li et al. 2012). Previous works mainly concentrated on data obtained by a single telescope (e.g. Melandri et al. 2008b; Klotz et al. 2009a; Cenko et al. 2009; Oates et al. 2009, 2011; Rykoff et al. 2009) and only a few authors compared the data from different instruments (e.g. Nysewander et al. 2009a; Kann et al. 2010b, 2011; Li et al. 2012; Liang et al. 2012). For example, Kann et al. (2010b, 2011) focused on the classification of the optical LCs and the host galaxy extinction. Li et al. (2012) andliang et al. (2012) concentrated on the optical LC shapes and particular features, as bumps, plateaus, late rebrightenings. Other works studied the dust extinction of the GRB host galaxies (e.g. Schady et al. 2012, 2010; Zafar et al. 2011) or the circumburst density profiles around GRB progenitors (Schulze et al. 2011). In this paper we analyze a large sample of 68 GRBs with optical and X-ray observations and known redshift, detected between December 2004 and December Our starting sample includes 165 GRBs with known redshift presented by Margutti et al (hereafter M 13). We collected the optical data from the literature and obtained well-sampled optical LCs for 68 GRBs from different telescopes and instruments. To compare the optical and X-ray observations, we used the X-ray data extracted and analyzed in M 13. We focused on the relationship between the optical and X-ray emission, comparing their restframe temporal and spectroscopic properties and their energetics. In particular, we investigated the forward-shock model and the synchrotron emission in the GRB afterglow. In Sect. 2 we detail the sample selection criteria, the data selection and reduction, the procedure followed for fitting the optical LCs and of the spectral energy distributions (SEDs). The results of our analysis are presented in Sect. 3 andarediscussedinsect.4. Themain conclusions are drawn in Sect. 5. We adopt standard values of the cosmological parameters: H 0 = 70 km s 1 Mpc 1, Ω M = 0.27, and Ω Λ = For the temporal and spectral energy index, α and β, we used the convention F ν (t,ν) t α ν β. Errors are given at 1σ confidence level unless otherwise stated. 2. Sample selection and data analysis We considered the 165 GRBs with known and secure redshift 2 observed by Swift/XRT between December 2004 and December 2010, presented in M 13. Among these GRBs, we selected those with optical observations and with optical data available in the literature. We used only the data from refereed papers and with more than five data points per filter. In this way we obtained a subsample of 68 long GRBs (Table C.9). This criterion automatically excluded short GRBs. The selection in spectroscopic 2 From Margutti et al. (2013) we used only optical spectroscopic redshifts and photometric redshifts for which we are able to exclude sources of degeneracy. We list the redshifts and luminosity distances of the GRBs of our sample in table5c.dat at CDS. redshift from the optical afterglow introduces a bias against highly absorbed optical afterglows (Fynbo et al. 2009; Perley et al. 2009a; Greiner et al. 2011). In fact, GRBs with optical spectroscopy have a substantially lower X-ray excess absorption and a substantially smaller fraction of dark bursts (Fynbo et al. 2009). On the other hand, our final aim is to compare X-ray and optical rest frame properties, and this can be carried out only with bright and well-sampled optical LCs. Within these constraints we collected a large number of data from more than one hundred telescopes with different instruments and filters (Table C.9). We analyzed the energetics and luminosities of these GRBs and calculated the SED in the optical/x-ray frequency range Optical data Magnitudes were converted into flux densities following standard practice (see Appendix A for details). For this analysis, we used only LCs that had more than five data points per filter and excluded upper limits. This is the best compromise between statistics (in the sense that we do not discard too many GRBs) and reliability (robust fit and energy measurement). All collected data will be available online 3. For each filter we fitted the optical LCs with the same fit functions as for the X-ray data in M 13. We chose these functional forms because they represent the optical LC shapes well and it facilitates comparing optical with X-ray data. We used optical data not corrected for reddening and these fit functions: 1. Single power-law: F ν (t) = Nt α. (1) 2. One or more smoothed broken power-laws: α ( ) 1,i α ( ) t s 2,i s i i t s i F ν (t) = N i + t i b,i t b,i (2) 3. Sum of power-law and smoothed broken power-law: ( ) α 2s ( ) α3s s F ν (t) = N 1 t α 1 t t + N 2 + t b, (3) t b where α is the power-law decay index, t b the break time, s the smoothness parameter (always fixed to 0.3, 0.5 or 0.8) and N the normalization. The best-fit parameters were determined using the IDL Levenberg-Marquard least-squares fit routine (MPFIT) supplied by Markwardt (2009) 4. The best-fitting function was chosen considering the χ 2 statistics. The bestfitting parameters are reported in Table C.1 5. The best-fit of the optical LCs and their residuals are shown in Figs. C.1C X-ray data The X-ray spectra were extracted using the method presented in M 13 (see also references therein). We fitted them with Xspec and the function tbabs*ztbabs*pow, which considers the hydrogen column density absorption of the Milky Way (N H,MW ) and of the host galaxy (N H,host ). The N H,MW was calculated with 3 The data will be available on the web site elenazaninoni.com The complete and machine-readable form of the table is provided at CDS (table1c.dat). A12, page 2 of 55

3 E. Zaninoni et al.: The gamma-ray burst optical light-curve zoo: comparison with the X-ray observations the nh tool, which uses the weighted average value from the Kalberla et al. (2005) map. The output data obtained from the X-ray spectrum are N H,X and the X-ray photon index 6 (Γ X ) (Table C.4 7 ) Optical/X-ray SEDs For each GRB, we created optical/x-ray SEDs at one or more epochs (Table C.8). The time intervals for the SEDs were chosen taking into account the shape of the X-ray and optical shapes: a) they belong to a determined phase of the X-ray LC that is steep decay, plateau or normal decay to avoid the X-ray LC breaks. In this way we obtained a SED both at early times (where the afterglow emission could be influenced by the prompt emission) and at late times (where the afterglow emission is very unlikely to be contaminated by the prompt emission); b) sometimes the SEDs were constructed during X-ray and optical flares. For the optical data, we did not extrapolate the optical LC, so that if for a given filter no data were available, the filter was excluded from the SED. For each filter with data in this time range, we calculated the flux density by integrating the optical LC over the considered time interval. We fitted the optical/x-ray SED accounting for absorption in the optical and X-ray ranges both locally (i.e., in the GRB host galaxy) and arising from the Milky Way (MW). For the optical band we used the extinction laws given by Pei (1992) (Eq. (20) and Table 4 therein) for the MW and the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC). For the X-ray data, we considered the model for the photoelectric cross section per HI-atom units for a given metallicity presented by Morrison & McCammon (1983), assuming solar metallicity. We first considered the case that the X-ray and the optical bands lie in the same spectral segment, hence the SED-fitting function is a combination of the absorption laws presented above and a single power-law: f ν (ν) = f 0 ν β op,x obs, (4) where ν obs is the observed frequency, β op,x the spectral index 6 and f 0 the normalization. From the input parameters, the Galactic hydrogen column density (N H,MW ), the Galactic reddening (E(B V) 8 MW ) and the redshift (z), we obtained the host galaxy hydrogen column density N H,op,X, reddening E(B V) host, and the spectral index β op,x. Then, we examined the hypothesis that the cooling frequency is between the optical and the X-ray bands and fit the data using the absorption laws plus a broken-power law: f ν (ν) = F 0 (ν β op obs step(ν obs,br ν obs ) +ν β X obs νβ X β op obs,br step(ν obs ν obs,br )), (5) where step is the step function, ν obs,br the observer frame break frequency between the optical and X-ray band, β op the optical spectral index, and β X the X-ray spectral index. The fit was performed in two ways: with β op let free to vary or fixed as β op = β X 0.5, as predicted theorically by Sari et al. (1998) and empirically by Zafar et al. (2011). Letting β op free to vary 6 The spectral index β is related to the photon index Γ by Γ=β The complete and machine-readable form of the table is provided at CDS (table4c.dat). 8 The E(B V) values were taken from NASA/IPAC Extragalactic Database (NED) website ( calculator.html), which uses the Schlegel et al. (1998) maps. Fig. 1. Cartoon representing the X-ray LCs types. For the X-ray LC shapes we used the code presented in M 13. Following the prescription of Bernardini et al. (2012a) and M 13, we denoted the different parts of the LCs as a) steep decay (S, green): first segment of type Ib and IIa LCs; the second segment of type IIb and III LCs; b) plateau (P, red): the first segment of type Ia LCs; the second segment of type Ib and IIa LCs; the third segment of type IIb and III LCs; c) normal decay (N, blue): type 0 LCs; the second segment of type Ia LCs; the third segment of type IIa LCs; the forth segment of type III LCs. did not lead to reliable results. Therefore the best-fit functions of the optical/x-ray SEDs may either be a single power-law or a broken power-law with β op = β X 0.5. To determine if a broken or a single power-law was required, we used an F-test probability <5% as threshold. The results of this selection are presented in Table C.8 and Figs. C.10C.18. The fit parameters are listed in Table C.2 9.InTableC.6 10 we list the optical data used for the SEDs. 3. Results 3.1. Spectral parameter distributions We considered the parameters obtained from fitting the optical/x-ray SEDs (β, N H, E(B V), ν BR ) with a single or broken power-law, as selected in Sect. 2.3 (Table C.8), and with a p-value 11 higher than We eliminated the results with errors larger than the data themselves and set to zero negative data that where consistent with 0 within the uncertainties. In total, 78% of our fits have p-value > 0.05 and 33 GRBs have more than one SED with a p-value > For the steep-decay SEDs, we 9 The complete and machine-readable forms of the tables are provided at CDS (table2c.dat, table3c.dat). 10 The complete and machine-readable form of the table is provided at CDS (table6c.dat). 11 The p-value is a number between zero and one and it is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. A12, page 3 of 55

4 Table 1. Characteristicquantities describing theparameter distributions (number of elements (#), mean (m), median (M), standard deviation (SD)), and best-fitting values from a Gaussian fit (mean (μ), standard deviation (σ)). Name # μ σ m SD M β op,x ± ± β X ± ± β op ± ± log (N H,BR /cm 2 ) ± ± log (N H,PL /cm 2 ) ± ± E(B V) MW ± ± E(B V) LMC E(B V) SMC A V,MW ± ± A V,LMC A V,SMC log (ν rest,br /Hz) log [(N H /cm 2 )/(A V /mag)] MW ± ± log [(N H /cm 2 )/(A V /mag)] LMC ± ± log [(N H /cm 2 )/(A V /mag)] SMC ± ± β S op,x β P op,x ± ± β N op,x ± ± β P X ± ± β N X ± ± β P op ± ± β N op ± ± log (N S H,PL /cm 2 ) log (N P H,PL /cm 2 ) ± ± log (N N H,PL /cm 2 ) ± ± log (N P H,BR /cm 2 ) ± ± log (N N H,BR /cm 2 ) ± ± log (ν P rest,br /Hz) log (ν N rest,br /Hz) log (L R,500 s /(erg s 1 )) ± ± log (L R,1 h /(erg s 1 )) ± ± log (L R,11 h /(erg s 1 )) ± ± log (L R,1 day /(erg s 1 )) ± ± log (F R, s /(erg cm 2 s 1 )) ± ± log (F 1keV, s /(erg cm 2 s 1 )) ± ± Notes. The subscript PL indicates the values obtained by fitting the SED with a single power-law, while BR indicates a broken power-law. S stands for steep decay, P for plateau, and N for normal decay. The values of E(B V) depend on the extinction model law used to fit the SED: Milky Way (MW), Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC). obtained a good fit (p-value > 0.05) with a single power-law for 9/13 SEDs and with a broken power-law for 1/13. We where unable to fit three SEDs. Accordingly, we did not consider parameters obtained with a broken power-law for the steep decay (i.e. ν rest,br, β op, β X, N H,BR ) in the distribution. For every distribution of the best-fitting values, we calculated the mean (m), the standard deviation (SD), and the median (M). When possible, we fitted the distributions with a Gaussian function obtaining the mean (μ) and the standard deviation (σ). All results are listed in Table 1. InFig.2 (top panels) we show the parameter distributions differentiating between the data obtained by fitting the SEDs with a single-power law (red, PL) or a broken-power law (blue, BR) and in Fig. 3 the distributions distiguishing the SED parameters extracted during the X-ray steepdecay phase (blue, S), the plateau (red, P) or the normal decay phase (gray, N) (see Fig. 1). We defined the X-ray LCs shapes as in M 13 (Fig. 1): 0 if there are no breaks, Ia or Ib if there is a break, IIa or IIb if there are two breaks, and III if there are three breaks. The differentiation between model Ia and Ib depends on the smoothness parameter s < 0ands > 0. Type IIa is the canonical shape (e.g. Nousek et al. 2006; Zhang et al. 2006), while type IIb starts with a shallow phase followed by a steep decay and then a normal decay. We presentin the bottomleftpaneloffig. 2 the distributions of host E(B V) for the MW, SMC and LMC for the SEDs for whichwewereabletodifferentiate between the extinction laws used Spectral index The mean spectral slope computed by fitting a single powerlaw is μ(β op,x ) = 0.95 ± This value is consistent with the spectral slope μ(β X ) = 0.97 ± 0.02 obtained using a broken A12, page 4 of 55

5 E. Zaninoni et al.: The gamma-ray burst optical light-curve zoo: comparison with the X-ray observations Fig. 2. Parameter distributions. The Colorcoding separates different SED best-fitting functions. Top panels. Blue: results obtained by fitting the SEDs with a broken power-law and the relative Gaussian fit (solid line). Red: results obtained by fitting the SEDs with a power-law and the relative Gaussian fit (solid line). a) The spectral indices (β) calculated fitting the SED with a single power-law (β op,x ) and with a broken power-law (β op, gray, and β X ). b) The hydrogen column density (N H ). Bottom panels. c) The optical extinction (E(B V)) distributions separated according to the different extinction laws: MW (blue), LMC (green), and SMC (orange). d) The rest frame break frequency (ν rest,br ) calculated by fitting the SEDs with a broken-power law. power-law; in fact the fit is largely weighted over the numerous X-ray data. The mean spectral slope of the optical part of the SED is μ(β op ) = 0.47 ± 0.02, computed by fixing β X β op = 0.5, hence it is simply a rigid shift of the distribution. The distributions of β computed over the three different parts of the X-ray LCs (steep decay phase, plateau, normal decay phase) have the following mean values (Figs. 3a,b,c): a) m(β S op,x ) = 0.85 (with SD = ), μ(β P op,x ) = 0.92 ± 0.03, μ(β N op,x ) = 0.99 ± 0.03; b) μ(βp X ) = 0.96 ± 0.02, μ(βn X ) = 1.00 ± 0.08; c) μ(β P op) = 0.49 ± 0.04, μ(β N op) = 0.47 ± From these distributions we note that the mean spectral index during the plateau is lower than during the normal decay phase, even though they are consistent within 2σ; in addition, the normal decay spectral index distribution is broaderthan during the plateau. Therefore we tested the evolution of β for each GRB (Fig. 4), with β = β op,x or β = β X depending of the fitting function used for each single SED (Table C.8). In most cases the spectrum becomes softer (22 GRBs, red lines), and only for ten GRBs it becomes harder (blue lines). For 26 GRBs we have only one valid SED fit (black dots). If we examine these relationships in the rest frame (inset), in particular only the plateau and normal decay data (magenta dots and orange squares) because we have 12 There are too few data to fit a Gaussian over the distribution. only few data for the steep-decay phase (light blue stars) and the unclassified phase (green triangles), then the relationships do not change Hydrogen column density and optical extinction The intrinsic hydrogen equivalent column density determines the X-ray absorption and measures the quantity of gas contained in the GRB host galaxy. The origin of this absorption is still debated, but is most likely due to absorption by the intergalactic medium, intervening absorbers or He in the HII region hosting the GRB (e.g. M 13, Campana et al. 2010, 2012; Behar et al. 2011; Watson et al. 2013). We calculated the intrinsic hydrogen equivalent column density after subtracting of the MW contribution, both by fitting the X-ray spectrum alone (N H,X ) and by a joined fit of optical and X-ray SED (N H,op,X ). N H,op,X was computed following the model presented by Morrison & McCammon (1983), which takes into account the photoelectric cross section per HI-atom units and for solar metallicity (Sect. 2.3). The N H values found with the two methods are consistent, as shown in Fig. 5, even the low values of N H (<10 21 cm 2 ) are consistent within two sigma. We therefore restricted our analysis to the intrinsic hydrogen equivalent column densities derived by the optical/x-ray A12, page 5 of 55

6 Fig. 3. Parameter distributions considering the X-ray LC part of the SED. Blue lines: steep-decay phase. Red lines: plateau. Gray lines: normal decay phase. a) β op,x : the spectral slopes calculated using a power-law as fitting function. b) β X and c) β op : the broken power law spectral slopes for the X-ray and optical data, respectively. d) N H,PL and e) N H,BR : the hydrogen column densities obtained using as SED fitting function a single power-law and a broken power-law, respectively. f) ν rest,br : the rest frame break frequency calculated fitting the SEDs with a broken-power law. SEDs (N H,op,X N H ). The distributions of the N H,op,X of the host galaxies derived from the single and broken power-law fits are consistent: μ(log (N H,PL /cm 2 )) = ± 0.11 and μ(log (N H,BR /cm 2 )) = ± 0.06 (Fig. 2b). As shown by the distributions of N H calculated in the different parts of the LCs (Figs. 3d,e), this parameter does not evolve with time because it has a similar mean value, within error, for the steep decay phase, the plateau and, normal decay phase. The values we found for the N H are consistent with those of M 13 within 1σ. In Fig. 2c we show the reddening distributions (E(B V)), differentiating between the best-fitting extinction laws (MW, LMC, SMC). The mean values are μ(e(b V) MW ) = 0.19 ± 0.02 mag, m(e(b V) LMC ) = 0.20 mag (SD = 0.16), m(e(b V) SMC ) = 0.20 mag, which corresponds to the host galaxy visual extinction (A V ) μ(a V,MW ) = 0.56 ± 0.10 mag, m(a V,LMC ) = 0.63 mag, m(a V,SMC ) = 0.59 mag 13. The mean A V,SMC is agrees with the value presented by Zafar et al. (2011). We studied the properties of the GRB host galaxy environment through the gas-to-dust ratio (N H /A V, Fig. 6). We considered N H,op,X and obtained for different extinction laws μ(log (N H /cm 2 )/(A V /mag)) MW ) = ± 0.05 (blue), μ(log ((N H /cm 2 ) /(A V /mag)) LMC ) = ± 0.08 (green) and μ(log ((N H /cm 2 )/(A V /mag)) SMC ) = ± 0.16 (orange). We compared these results with the N H /A V values available 13 A V = R V E(B V) with R MW V (Pei 1992). = 3.08, R LMC V = 3.16 and R SMC V = 2.93 A12, page 6 of 55

7 E. Zaninoni et al.: The gamma-ray burst optical light-curve zoo: comparison with the X-ray observations Fig. 4. Evolution of β with time for individual GRBs. For every GRB we considered the correct spectral index as selected in Table C.8, hence β can be β op,x or β X depending on the chosen SED fitting function, a single power-law or a broken power-law. Blue dotted lines: the initial spectral slope is steeper than the final spectral slope. Red dotted lines: the initial spectral slope is flatter than the final spectral slope. Light blue stars: steep decay data. Magenta dots: plateau data. Orange squares: normal decay data. Black: only one SED is available for these GRBs and precisely during the steep decay (stars), the plateau (dots), and normal decay (squares). Inset: the same as the principal plot, but in the rest frame. Fig. 5. Comparison between the N H calculated from the X-ray spectrum (N H,X ) and the optical/x-ray SED (N H,op,X ). Blue triangles stand for the broken power-law fit function and black dots for the simple power-law. Red line: N H,X = N H,op,X. in the literature for the MW, LMC and SMC: log ((N H /cm 2 )/ (A V /mag)) MW = (Fig. 6, blue star; Bohlin et al. 1978), log ((N H /cm 2 ) / (A V /mag)) LMC = (Weingartner & Draine 2001) andlog((n H /cm 2 )/(A V /mag)) SMC = (Martin et al. 1989). To compare the Magellanic Clouds data of the N H from the literature with our results, calculated assuming solar abundances, we converted the values from the literature assuming a metallicity Z = 0.26 Z for the LMC and Z = 0.14 Z for the SMC (Draine 2003 and references therein). We obtained log ((N H /cm 2 )/(A V /mag)) LMC = and log ((N H /cm 2 )/(A V /mag)) SMC = (Fig. 6, green and orange stars, respectively). Our analysis shows that the gas-to-dust ratios of GRBs are higher than the values calculated for the MW, the LMC, and SMC assuming sub solar abundances (e.g. Schady et al. 2010, 2012). We caution, however, that our distributions characterize GRBs that are not heavily absorbed in the X-rays and in the optical band, because our sample is redshift-selected Break frequency The rest frame break frequency distribution has a peak around log (ν rest,br /Hz) 16 (Fig. 2d). The values are spread between the optical and the X-ray band frequencies and it is not possible to fit a Gaussian function to these data. Because most of our data where taken at late times, they probably correspond to a slow cooling regime for a homogeneous medium, when the break-frequency evolves as t 1/2 (Sari et al. 1998), moving from the X-ray toward the optical frequencies. Since we cannot follow these changes for a single burst, we tested whether we could find any correlation between the mean time at which we measure the break frequency and the break frequency itself. If GRBs had a similar behavior, the time and the break frequency would be correlated. Figure 7 shows that the peak at low frequencies is spread over a long time interval, and Fig. 6. Distribution of log ((N H /cm 2 )/(A V /mag)) considering the three different extinction laws used: MW (blue), LMC (green), and SMC (orange). Stars: reference values of the ratios N H /A V from the literature. Blue star: log ((N H /cm 2 )/(A V /mag)) MW = (Bohlin et al. 1978). Green star: log ((N H /cm 2 )/(A V )/mag)) LMC = (assuming sub solar abundances). Orange star: log ((N H /cm 2 )/(A V /mag)) SMC = (assuming sub solar abundances). there is no evidence of a correlation between time and frequency (Fig. 7, left). This may be due above all to the dependence of the the break frequency on other parameters, in part to the uncertainties in its measurement, and also because we considered data from different GRBs Luminosity and energetics In Fig. 8 we plot the X-ray (1 kev, gray lines) and optical (R band) blue lines) rest-frame LCs of the GRBs in our sample 14. The optical and X-ray LCs have similar luminosities; our redshift-selected sample favors bright optical GRBs. 14 The X-ray and optical data are k-corrected. Optical data are not corrected for Galactic and host galaxy absorption; X-ray data are corrected for Galactic and host galaxy absorption. Therefore, the optical luminosity derived is a lower limit of the real value. However, since the GRBs considered have smaller absorption (see Sect ) the real luminosity is about a factor 2 higher than the one considered. A12, page 7 of 55

8 Fig. 7. Break frequency. Left: break frequency (ν rest,br ) vs. the mean time (t rest,m ) of the interval in which the SED is calculated. Right: the distribution of the break frequencies. Blue: t rest,m < 500 s. Red: 500 < t rest,m < 10 4 s. Gray: 10 4 < t rest,m < 10 5 s. Orange: t rest,m > 10 5 s. The time intervals have been arbitrarily chosen. Fig. 8. X-ray (1 kev, gray) and optical (R band, blue) LCs in the rest frame. For the GRBs in our sample, we compared the optical (R band) and X-ray (at 1 kev) flux (Fig. 9) inacommon rest frame time interval ( s): the X-ray emission (log (μ(f X )/(erg cm 2 s 1 )) = 12.54, σ = 0.49) is on average one order of magnitude fainter than the optical (log (μ(f op )/(erg cm 2 s 1 )) = 11.41, σ = 0.34). The optical LCs can show an early-time rise or a quasiconstant phase (optical plateau), followed by a decay. Panaitescu & Vestrand (2011) claimed that there are some correlations involving the energies and luminosities calculated at the peak of the early-time rise or at the end of the optical plateau, which are predicted by theoretical models. We verified these relations in the observer and the rest frames considering the GRBs with an optical LC with an initial peak and with an initial optical plateau (Table 3). To compute the relations between two parameters, we used the best-fitting procedure, which accounts for the sample variance (D Agostini 2005). All results are listed in Table 2 and presented in Figs We confirm the correlation between the optical energy (L t rest with L = L end, L pk ) and the isotropic gamma-ray energy We used the values presented in M 13. Fig. 9. Energetics. Distribution of the X-ray (1 kev, red, solid line) and optical (R band, blue, solid line) flux calculated in a common rest frame time interval ( s) for our sample and their distributions (red, dotted line and blue, dashed line, respectively). (E γ,iso, Amati et al. 2008) and the optical energy and the energy calculated in the BAT energy band 15 (Eγ ;Fig.10). However, for all these correlations the data show very broad distributions. There is a weak indication that the optical plateau end luminosities and the relative observer and rest frame times are correlated (Fig. 11, left). The same occurs for the peak luminosities and the relative observer and rest frame times (Fig. 11, right). Since there are few elements in our sample, the correlation is not reliable. In the observer frame, the peak flux correlates with the peak time (Fig. 12, bottom), but the optical plateau end fluxes and their times are not related (Fig. 12, top). We had only few data for this as well: 19 GRBs for the peak relation and 14 for the plateau, and a few discrepant cases. We also measured the optical luminosity distributions at four different rest frame times: 500 s, 1 h, 11 h, and 1 day. The 11-h time frame is commonly chosen because it is reasonable to assume that the cooling frequency has not passed the optical band yet (Freedman & Waxman 2001; Piran et al. 2001), in addition, 11 h and 1 day are approximately the A12, page 8 of 55

9 E. Zaninoni et al.: The gamma-ray burst optical light-curve zoo: comparison with the X-ray observations Table 2. Two-parameter correlations involving optical luminosities and fluxes. X Y m q σ ρ K R L t rest Eγ ± ± ± L t rest E γ,iso 0.78 ± ± ± t rest,pl L end,pl 0.83 ± ± ± t obs,pl L end,pl 0.77 ± ± ± t rest,pk L end,pk 1.40 ± ± ± t obs,pk L end,pk 1.32 ± ± ± t obs,pl F PL 0.48 ± ± ± t obs,pk F PK 0.93 ± ± ± Notes. From left to right: X and Y parameters to be correlated (the best-fitting law reads log (Y) = q+m log (X)); best-fitting parameters as obtained accounting for the sample variance (D Agostini 2005): slope (m),normalization (q), intrinsic scatter (σ); errors are given at 95% c.l. The last three columns listthe value of the Spearman rank (ρ), Kendall coefficient K, and R-index r statistics. Fig. 10. Relations between the optical energy (time luminosity) of the optical plateau end (blue dots) and of the peak (black dots) and the kev BAT energy (top) and the isotropic prompt emission energy (bottom). Dashed line: best-fitting power-law model obtained accounting for the sample variance (D Agostini 2005). Gray area: the 68% confidence region around the best fit. The results of the fit are listed in Table 2. times at which several authors found a bimodal distribution 16 of the luminosities (Liang & Zhang 2006; Nardini et al. 2006; Kann et al. 2006; Nardini et al. 2008). We found no bimodal distribution in our data, as asserted in recent studies (Melandri et al. 2008b; Oates et al. 2009, 2011; Kann et al. 2010b, 2011). The mean luminosity simply decreases with time (Fig. 13): μ(log (L 500 s /erg s 1 )) = ± 0.06 (64 GRBs), μ(log (L 1h /erg s 1 )) = ± 0.06 (57 GRBs), μ(log (L 11 h /erg s 1 )) = ± 0.07 (40 GRBs) and μ(log (L 1day /erg s 1 )) = ± 0.09 (32 GRBs) (Table 1). 16 Possibly caused by a bimodality in the optical luminosity function or by the absorption of gray dust in a fraction of bursts (Nardinietal. 2008) The optical LCs The optical LCs show different shapes and features (Table 3;see also Li et al. 2012, their Fig. 2). For each GRB, we selected the optical LC observed with the filter with the widest temporal coverage and the most reliable fit (Table C.1 17 ). In general, they have a rising or constant part, which can occur at any time. This occurs for 53 GRBs in our sample. Only ten GRBs show a simple power-law trend and five GRBs have an LC with an initial decay followed by an almost constant optical flux. The optical LCs with a single power-law decay have an initial time >600 s in the observer frame (GRB , GRB , GRB A) or they are poorly sampled (GRB , GRB , GRB , and GRB ). GRB and GRB C show weak variability in their optical LCs, even though their best-fit function is a simple power-law. GRB A has a well-sampled optical LC, fitted with a single power-law. Of the five GRBs with an initial decay followed by almost constant flux, GRB and GRB show a shallow phase at late times ( 10 6 ), which may be due to the host galaxy. For GRB , GRB , and GRB the LC break occurs at s (observer frame). For these five GRBs the initial time of the optical observations is 100 s. Even though these GRBs do not show variable LCs, we do not know what happened before the observations; they could have a very variable LC similar to GRB B (Racusin et al. 2008). There are 14 GRBs with an optical LC with an early peak (i.e., an initial rise followed by a decay) and 14 with a quasiconstant phase (i.e., optical plateaus, an initial quasi-constant phase followed by a decay). The optical plateaus and rises in the LCs are interpreted as the onset of the forward shock emission, when the blast wave decelerates: the peaked LCs correspond to an impulsive ejecta release, where all ejecta have the same Lorentz factor after the burst phase; the plateaus are caused by the energy injection in the forward-shock due to an extended ejecta release, a wide distribution of the ejecta initial Lorentz factors or both (e.g. Panaitescu & Vestrand 2011; Oates et al. 2009), or the onset of the afterglow for the wind medium (e.g. Chevalier & Li 1999; Ghirlanda et al. 2012). Sixteen GRBs show a late-time re-brightnening (i.e., at late times the LC displays a rise phase followed by a decay phase 17 The complete and machine-readable form of the table is provided at CDS (table1c.dat). 18 In this case we consider the overall trend, since it is difficult to fit the LC with a more complicated fit function. For a detailed study of this LC see Sollerman et al. (2007). A12, page 9 of 55

10 Fig. 11. Relations between the optical luminosity of the end of the plateau (left) and of the peak (right) and the relative observer (top) andrest (bottom) frame time. Dashed line: best-fitting power-law model obtained accounting for the sample variance (D Agostini 2005). Gray area: the 68% confidence region around the best fit. The results of the fit are listed in Table 2. with the same slope as before). The re-brightening may be related to the jet structure, and seems to agree with the on-axis two-component jet model, with the re-brightening corresponding to the emergence of the slow component (e.g. Jakobsson et al. 2004; Racusin et al. 2008; Liang et al. 2012). Three GRBs show a series of initial large bumps (i.e., more than one peak). GRB B shows two bumps during the X-ray plateau and a shallow decay starting roughly at the beginning of the X-ray normal decay. The two optical peaks are not correlated with the high-energy emission and the subsequent optical bump is assumed to trace the onset of the forward shock (Rykoff et al. 2009). The optical LC of GRB has two bumps that coincide with the X-ray plateau. These bumps could be associated to a change of the circumburst density (Lazzati et al. 2002; Cenko et al. 2009). The GRB optical LC was modeled by multiple energy injections into the forward shock, and not with the central engine, since the fluctuations occur on a long timescale (Rossi et al. 2011). The first peak is assumed to be the onset of the afterglow, while the following two bumps are produced by the central engine activity (Rossi et al. 2011). Five optical LCs show small bumps (i.e., weak fluctuations over the power-law decay). The optical bumps could be related to the erratic late-time central engine activity (Li et al. 2012). The optical LCs have a complex behavior, and during a welldefined X-ray LC phase the optical LC can rise and then decay, or vice versa. Specifically 19 : Steep decay: 42% of the optical LCs rise, 32% decay, 16% are constant, and 10% have a complex behavior (rise-decay, bumps). 19 The percentage refers to single LC parts, not to the total number of GRBs. Fig. 12. Relations between the optical flux and the observer time of the end of the plateau (top) and the peak (bottom). Dashed line: bestfitting power-law model obtained accounting for the sample variance (D Agostini 2005). Gray area: the 68% confidence region around the best fit. The results of the fit are listed in Table 2. Plateau: 16% of the optical LCs rise, 47% decay, 8% are constant, 26% rise and decay and 4% have one or more bumps. A12, page 10 of 55

11 E. Zaninoni et al.: The gamma-ray burst optical light-curve zoo: comparison with the X-ray observations Table 3. Subdivision of the GRBs in our sample according to the optical LC features. Initial peak a A a A a A a C A a A Initial shallow phase A A Late-time re-brightening A A A A Series of initial large bumps B Small bumps b A b Notes. (a) At late times. (b) For the R C observations. Fig. 13. Distribution of the optical R luminosity calculated for four different rest frame times: 500 s, 1 h, 11 h, and 1 day. The black solid line corresponds to the Gaussian fit of the data. The results are listed in Table 1. Normal decay: 77% of the optical LCs decay and 23% rise and decay or have a more complex behavior. The complexity of the LCs decreases as a function of time Comparison between the optical and X-ray LCs For every GRB we compared the optical LC slopes with the contemporaneous X-ray LC slopes. For both the optical and the X-ray LCs, we considered only the continuum part of the LC, excluding small bumps and flares (see M 13). As in the previous section, for each GRB, we selected the optical LC observed with the filter with the widest temporal coverage and the most reliable fit (Table C.1 20 ); the X-ray LC parameters are those derived in M 13. From the synchrotron spectrum (Sari et al. 1998), if ν c < ν op < ν X, with ν c the cooling frequency, the difference between the contemporaneous optical and X-ray slopes is Δα = 0. If ν op < ν c < ν X, for the slow cooling regime, Δα = ±1/4 21. We subdivided our sample into three groups, depending on whether the pairs of the optical/x-ray slopes follow the relation Δα = 0, 1/4, or not at all within 1σ 22 (Table 4 and Fig. 14): Group A: all pairs of slopes of the same GRB satisfy the relation Δα = 0, ±1/4 (13 GRBs). Group B: some slopes of the same GRB satisfy the relation Δα = 0, ±1/4 (27 GRBs). Group C: no slopes of the same GRB satisfy the relation Δα = 0, ±1/4 (28 GRBs). Some X-ray LCs show an initial steep decay; this is generally not present in the optical LCs, which display a rise, a plateau, or a normal decay. The X-ray steep decay is well explained as the decay of the prompt emission, and its slope value is particularly sensitive to the chosen zero time of the power-law decay, t 0 0 (e.g. the BAT trigger time) or t 0 = t 90 (for details see M 13). For this reason, the steep-decay phase was not considered in our classification. We plot α X vs. α op for every GRB in Fig. 15 (e.g. Urata et al. 2007). About half of the α X vs. α op couples refer to the X-ray LC normal decay phase and the other half to the plateau. We noted 20 The complete and machine-readable form of the table is provided at CDS (table1c.dat). 21 This is valid in the slow cooling regime for the constant interstellar medium (ISM) and the wind case: for ν m <ν<ν c, α 1 = 3(p 1)/4, and for ν>ν c, α 2 = (3p 2)/4 in the ISM case, so α 1 α 2 = (3p 3 3p + 2)/4 = 1/4; for the wind case, for ν m <ν<ν c, α 1 = (3p 1)/4, and for ν>ν c, α 2 = (3p 2)/4, so α 1 α 2 = (3p 1 3p + 2)/4 = 1/4. 22 A similar method was used by Panaitescu & Vestrand (2011) toclassify coupled and decoupled LCs. A12, page 11 of 55

12 Table 4. List of GRBs in the three groups. GRB Group A (13 GRBs) A A B A A A Group B (27 GRBs) B A A A B A B A A Group C (28 GRBs) A C A A C Notes. See Appendix B for more details. that the GRBs in Group A and B have more complex LCs than the Group C GRBs, indeed most of the X-ray and optical LCs of Group A and B GRBs have Type IIa or III shapes. Therefore, when GRBs have LCs that are well sampled and have a good time coverage, hence with more complicated shapes, the X-ray and the optical LCs show a similar trend. When we have fewer data, we cannot compare some parts of the LCs and perhaps the observed slope is different from the real behavior of the LC. X-ray flares do not influence the relation between the X-ray and optical LCs, because in Group A there are 5/14 GRBs with flares (36%), in Group B there are 8/26 (31%), and in Group C are 8/28 (29%). This agrees with the percentage found in other samples (Chincarini et al. 2010a; Margutti et al. 2013). 4. Discussion We presented the analysis of a large and homogeneous data set, useful for studying the GRB rest frame properties and for comparing the optical and X-ray emission. The comparison between the X-ray and the optical LCs and SEDs enables us to investigate the nature of their emission mechanism and to verify if they have the same origin. For the internalexternal shock model (Sari et al. 1998), the forward shock propagating into the external medium gives rise to the X-ray and optical emission. If the optical and the X-ray LCs have similar shapes and slopes, they could be caused by synchrotron emission and probable are produced by the forward shock (e.g. Zhang et al. 2006). Indeed, the X-ray emission is mainly influenced by the central engine activity: the steep decay is thought to be the tail of the prompt emission (Kumar & Panaitescu 2000) oritis direct emission from the central engine (Barniol Duran & Kumar 2009). The plateau reflects the effect of energy injection into the forward shock (e.g. Zhang et al. 2006). The optical LCs show various features: initial peaks or constant phases, which are probably caused by the onset of the forward shock (e.g. Panaitescu & Vestrand 2011), late-time rebrightenings that may depend on the structure of the jet (e.g. Racusin et al. 2008), and small bumps linked to the central engine activity (Li et al. 2012). Fig. 14. Comparison between optical and X-ray LCs, examples of the GRBs in each group. Group A: GRB (blue/light blue). Group B: GRB (red/orange). Group C: GRB A (purple/magenta). For each panel: Top. Colored points: X-ray data. The data in light color and bright color represent the continuum and the flaring portions, respectively, as calculated by M 13. Gray dashed lines: X-ray break times. Gray points: optical data. Black solid line: fit to the data. Gray solid lines: components of the fit function used to fit the optical data. Hashed gray boxes: SED time intervals. Middle. Ratio between the optical data and their fit function. Bottom. Ratio between the fit to the X-ray continuum and the optical LC. See Figs. C.1C.9 for the other GRBs of our sample. Thanks to our sample of LCs and SEDs, we were able to discuss the similarities and differences of the optical and X-ray emission by comparing their LCs (Sect. 4.1). In Sect. 4.2 we considered the forward-shock model and the closure relations (e.g. Sari et al. 1998; Zhang et al. 2006), and in Sect. 4.3 we presented the radio/optical/x-ray SED of GRB , which is A12, page 12 of 55

13 E. Zaninoni et al.: The gamma-ray burst optical light-curve zoo: comparison with the X-ray observations Fig. 15. Comparison between the X-ray LC slope (α X ) and the optical one (α op ). Red (blue) dots: data for the plateau (normal decay) phase that agree with the Δα = 0, 1/4 relation whitin the 1σ errors. Gray dots: the data that do not follow the Δα = 0, 1/4 relation. Red solid line: Δα = 0. Gray dashed lines: Δα = ±1/4. well fitted by the synchrotron spectrum. Finally, we investigated the role of the optical emission in the three-parameter correlation between E X,iso E γ,iso -E pk (M 13, Bernardini et al. 2012b) LC phases Steep-decay phase, the plateau, and the X-ray flares The steep-decay phase of the X-ray LCs is either the highlatitude emission after the end of the prompt emission (Kumar & Panaitescu 2000), or it may be part of the prompt emission itself, as proposed by Barniol Duran & Kumar (2009)andKumar et al. (2008), since the X-ray flux is smoothly connected to the γ-ray emission (Tagliaferri et al. 2005b; Goad et al. 2006) and is characterized by a strong hard-to-soft spectral evolution (Butler & Kocevski 2007). In this phase most of the optical LCs rise (42%) or have a complex behavior, with bumps, peaks, and plateaus (26%), which makes their slopes different from the X-ray steep-decay slopes. The remaining 32% decay during this phase, but of these there are only two cases whose optical LC slopes are similar to the X-ray slopes (e.g. GRB , GRB A). In some cases, there are X-ray flares superimposed on the steep decay, which are linked to the central engine activity (e.g. Zhang et al. 2006; Chincarini et al. 2010a). The X-ray plateau is interpreted as an injection of energy into the forward shock (e.g. Zhang et al. 2006) and agrees with this prediction since there is no significant spectral evolution (Bernardini et al. 2012a). The source of the energy injection could be the power emitted by a spinning-down newly-born magnetar (Dai & Lu 1998; Zhang & Mészáros 2001; Corsi & Mészáros 2009) that refreshes the forward shock (Dall Osso et al. 2011) or the fall-back and accretion of the stellar envelope on the central black hole (Kumar et al. 2008). During this phase the optical LCs behave in different ways and 46% of them rise or show peaks or bumps. From the comparison of the X-ray and optical LCs, we noted that there is a relation between the occurrence of the X-ray flares and the peak time of the optical LCs (Figs. 1618). That is, when the flares are observed during the X-ray steep decay phase, the optical peak occurs early and before the beginning of the X-ray plateau, while if there are no flares or late-time flares, the optical peak occurs during the X-ray plateau. The peak of the optical LC occurs during or at the end of the steep-decay phase if there are X-ray flares in this phase, as in GRB , GRB , GRB , GRB , GRB , GRB , GRB , GRB , GRB A, and GRB If the X-ray flares occur during the X-ray plateau, the optical LC peak or the end of the optical plateau occurs during the X-ray plateau: GRB , GRB , GRB A, GRB , GRB B, and GRB In some cases it is difficult to evaluate this relation between the X-ray flares and the optical LCs. For GRB the computation of the X-ray break time is influenced by the presence of a very bright X-ray flare (e.g. Margutti et al. 2010b, for details). Accordingly, if we consider a break time 200 s, the peak of the optical LC would be synchronous with the end of the steep-decay phase, and in this case it would be part of the group of GRBs with the break of the optical LC occurring during or at end of the steep decay. The GRB v filter data show a variability that corresponds with the X-ray flare, even if there is no true break. GRB has a Type 0 X-ray LC with a superimposed flare that temporally corresponds to the optical peak. At late time it shows another optical re-brightening that corresponds to a weak X-ray flux variation. GRB A belongs to the 17/437 Type IIb GRBs with complete LCs (M 13). The end of the X-ray steep decay corresponds to the peak of the optical LC even though there are no flares during the steep decay. Unlike, the X-ray LC is not well sampled after the steep decay phase. If there are no flares, the optical break occurs during the X-ray plateau. GRB A and GRB A simply follow the trend of the X-ray data. The LC of GRB is similar to that of GRB A and GRB A, but there are no data during the X-ray steep decay. For GRB , GRB A, GRB , GRB , GRB , and GRB some data are lacking during the plateau and the steep decay is not well sampled or is of short duration. GRB C and GRB B have Type Ia X-ray LC (M 13) and so there is no steep decay phase. GRB has no X-ray data before the optical peak. For GRB we have only late-time data (t start,x s). The optical peak of GRB A corresponds to the end time of the X-ray plateau, but there are no X-ray data during the plateau. GRB , whose observations started 2000 s after the trigger, has a Type 0 LC and shows an optical break at about s, even though the X-ray LC does not show flux variations. For GRB , GRB , GRB A, GRB A, GRB , GRB , GRB A, GRB , GRB , GRB , GRB , GRB A, GRB , GRB , and GRB the X-ray data are very poor. The relation between the X-flares and the optical peaks and plateaus is displayed also in GRB , GRB A, GRB , GRB , and GRB A studied by Li et al. (2012)andLiang et al. (2012) Normal decay phase The X-ray normal decay is present in most of the X-ray LCs of our sample (63/68). Seventy-seven per cent of the optical LCs decay during the X-ray normal phase, but only 62% have a similar slope. In addition, only a few cases follow the closure relations. A12, page 13 of 55